The Spanish champions were lacklustre for long spells of the first half at San Mames and fell behind to a Mikel Rico header after 17 minutes.
However, Simeone saw his side turn in a much improved display after the break as former Real Sociedad man Griezmann levelled within a minute of the second half.
Raul Garcia's penalty gave the visitors the lead before Griezmann continued his impressive record against his former club's rivals by registering a hat-trick to keep Atletico on the tails of Barcelona and Real Madrid going into the mid-season break.
Atletico sit four points behind leaders Real, who have a game in hand.
Earlier, Villarreal continued their fine form with a comfortable 3-0 win over Deportivo La Coruna - despite seeing Pina sent off 10 minutes from time.
Marcelino's side have now won their past five league matches in a row and have not conceded in La Liga since November 23.
Jonathan dos Santos got the hosts off to a strong start at the Madrigal 10 minutes in before Luciano Vietto struck twice in the space of six second-half minutes.
Pina's late red mattered little as Deportivo remain winless on the road in the league since mid-September.
Granada were on course to end 2014 out of the relegation spots but were held to a 1-1 draw by Getafe - Emiliano Velazquez equalising 12 minutes from the end.
Joaquin Caparros' side slipped to 18th after Almeria's 1-0 win over Celta Vigo on Friday but would have moved out of the bottom three again had Jhon Cordoba's opener proven a winner.
However, Velazquez had other ideas - with fellow strugglers Elche also staying in the drop zone following a 2-1 defeat to Malaga.
The league's bottom side were on course to join Almeria, Depor and Granada on 13 points when David Lomban opened the scoring on the stroke of half-time at the Estadio Manuel Martinez Valero.
Ignacio Camacho struck early in the second half before Luis Alberto completed the turnaround for Malaga who move to 30 points - a club-record haul at this stage of a La Liga season.